In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity
$ \begin{eqnarray*} \left\{ \begin{array}{lll} (-\Delta)^su = \lambda u+|u|^{p-2}u+|u|^{2^\ast_s-2}u, & x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}u^2{\rm d}x = a^2, \ u\in H^{s}(\mathbb{R}^N), \end{array}\right. \end{eqnarray*} $
where $ N\geq2 $, $ s\in(0, 1) $, $ a > 0 $, $ 2 < p < 2^\ast_s\triangleq\frac{2N}{N-2s} $ and $ (-\Delta)^s $ is the fractional Laplace operator. In the purely $ L^2 $-subcritical perturbation case $ 2 < p < 2+\frac{4s}{N} $, we prove the existence of a second normalized solution under some conditions on $ a $, $ p $, $ s $, and $ N $. This is a continuation of our previous work (Z. Angew. Math. Phys., 73 (2022) 149) where only one solution is obtained.
Citation: Xizheng Sun, Zhiqing Han. Note on normalized solutions to a kind of fractional Schrödinger equation with a critical nonlinearity[J]. AIMS Mathematics, 2024, 9(8): 21641-21655. doi: 10.3934/math.20241052
In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity
$ \begin{eqnarray*} \left\{ \begin{array}{lll} (-\Delta)^su = \lambda u+|u|^{p-2}u+|u|^{2^\ast_s-2}u, & x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}u^2{\rm d}x = a^2, \ u\in H^{s}(\mathbb{R}^N), \end{array}\right. \end{eqnarray*} $
where $ N\geq2 $, $ s\in(0, 1) $, $ a > 0 $, $ 2 < p < 2^\ast_s\triangleq\frac{2N}{N-2s} $ and $ (-\Delta)^s $ is the fractional Laplace operator. In the purely $ L^2 $-subcritical perturbation case $ 2 < p < 2+\frac{4s}{N} $, we prove the existence of a second normalized solution under some conditions on $ a $, $ p $, $ s $, and $ N $. This is a continuation of our previous work (Z. Angew. Math. Phys., 73 (2022) 149) where only one solution is obtained.
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Xizheng Sun, Zhiqing Han. Note on normalized solutions to a kind of fractional Schrödinger equation with a critical nonlinearity[J]. AIMS Mathematics, 2024, 9(8): 21641-21655. doi: 10.3934/math.20241052
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